# Symmetric and Skew Symmetric Matrices

## Symmetric and Skew Symmetric Matrices

Symmetric Matrix: A square matrix A is said to be a symmetric matrix if AT = A.

Examples:

1. $$A=\left[ \begin{matrix}2 & 3 & 1\\3 & 4 & 5\\1 & 5 & 7\\\end{matrix} \right]$$.
$${{A}^{T}}=\left[ \begin{matrix}2 & 3 & 1\\3& 4 & 5\\1 & 5 & 7\\\end{matrix} \right]$$.

Hence A = AT then A is symmetric matrices.

$$A=\left[ \begin{matrix}1 & -1\\-1 & 2\\\end{matrix} \right]$$.
$${{A}^{T}}=\left[ \begin{matrix}1 & -1\\-1 & 2\\\end{matrix} \right]$$.

Hence A = AT then A is symmetric matrices.

3. If A is a square matrix then show that A + AT, AAT are symmetric matrices.

Solution:

(A + AT)T

= AT + (AT)T

= AT + A

= A + AT

→ A + AT is a symmetric matrix.

(AAT)T = (AT)T

AT = AAT → AAT is symmetric matrix.

Note: If A is a symmetric matrix and k is a scalar then kA is a symmetric matrix.

Skew Symmetric Matrix: A square matrix A is said to be a skew symmetric matrix if AT = -A.

Examples:

1. $$A=\left[ \begin{matrix}0 & 1 & -2\\-1 & 0 & 3\\2 & -3 & 0\\\end{matrix} \right]$$.

$${{A}^{T}}=\left[ \begin{matrix}0 & -1 & 2\\1 & 0 & -3\\-2 & -3 & 0\\\end{matrix} \right]$$.

$$-{{A}^{T}}=-\left[ \begin{matrix}0 & -1 & 2\\1 & 0 & -3\\-2 & -3 & 0\\\end{matrix} \right]$$.

$$-{{A}^{T}}=\left[ \begin{matrix}0 & 1 & -2\\-1 & 0 & 3\\2 & 3 & 0\\\end{matrix} \right]$$.

Hence A=-A T , then A is skew symmetric matrices.

$$A=\left[ \begin{matrix}0 & 5\\-5 & 0\\\end{matrix} \right]$$.

$${{A}^{T}}=\left[ \begin{matrix}0 & -5\\5 & 0\\\end{matrix} \right]$$.

$$-{{A}^{T}}=-\left[ \begin{matrix}0 & -5\\5 & 0\\\end{matrix} \right]$$.

$$-{{A}^{T}}=\left[ \begin{matrix}0 & 5\\-5 & 0\\\end{matrix} \right]$$.

Hence A = A T, then A is are skew symmetric matrices.

3. If A is a square matrix then show that A – AT is an skew symmetric matrix.

Solution:

(A – AT)T = AT

= – (AT)T

= AT

= – A

= -(A – AT) → A – AT is an skew symmetric matrix.

Note: If A is an skew symmetric matrix and k is a scalar then kA is an skew symmetric matrix.

Note: A skew symmetric matrix of order 3 is of the form

$$\left[ \begin{matrix}0 & a & b\\-a & 0 & c\\-b & -c & 0\\\end{matrix} \right]$$.